1. Chapter 3.1 Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)  Introduction The solution for Stokes waves is valid in deep or intermediate water depth. It is assumed that the wave steepness is much smaller than one.

2.  Nondimensional Variables

3. Nondimensional Governing Equation & Boundary Conditions

4.  Perturbation (Stokes Expansion)

5.  Hierachy Equations

9. Solving the non-dimensional Equations from lower order (j=1) to higher order (j=3) for the non-dimensional solutions (wave advances in the x-direction).

11. The non-dimensional solutions are then transferred back to the dimensional form.

14. Convergence For the fast convergence of the perturbed coefficient,, must be much smaller than unity, which is consistent with weakly nonlinear assumption. However, when the ratio of depth to wave length is small, the Stokes perturbation may not be valid.

16. A few striking features of a nonlinear wave train can be described for the above equation: The crests are steeper and troughs are flatter; (see applet (Nonlinear Wave Surface)). Phase velocity increases with the increase in wave steepness. Non-closed trajectories of particles movement. (see applet (N-Trajectory)). Nonlinear wave characteristics (up to 2 nd order).

17. Wave advancing in the x-direction

18. Particle Trajectory

24. Dynamic Pressure

25. Radiation Stress Radiation stress: defined as the time average of excess quasi momentum flux due to the presence of a periodic wave train.